How does $r$ for univariate regression relate to the general definition of $R^2$?

Question

We know that the correlation coefficient in a univariate regression case between $x$ and $y$ is

$$r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x - \bar{x})^2\sum_{i=1}^n (y - \bar{y})^2}} \\ r^2 = \left( \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x - \bar{x})^2\sum_{i=1}^n (y - \bar{y})^2}} \right)^2 $$

The general definition for $R^2$ is \begin{align} R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \\ SS_{res} = \sum_i (y_i - \hat{y}_i)^2 \\ SS_{tot} = \sum_i (y_i-\bar{y})^2 \\ \\ SS_{reg} = \sum_i (\hat{y_i}-\bar{y})^2 \\ \\ \end{align}

In the univariate regression form, $r^2$ isn't a function of the $\hat{y}$, and the latter $R^2$ isn't a function of $x$. How is $r^2 = R^2$?

Answer 1

Recall that $$ \hat{\beta} = \frac{\sum( x_i - \bar{x} )( y_i - \bar{y} ) }{ \sum (x_i - \bar{x} ) ^2}. $$ Since $SSreg = \hat{\beta} ^ 2 \sum (x_i - \bar{x} ) ^2$,

\begin{align} R ^ 2 &= \frac{SSreg}{SST}\\ & = \frac{\hat{\beta}^2 \sum (x_i - \bar{x} ) ^2}{\sum ( y _i - \bar{y})^2}\\ & = \frac{ \left( \sum( x_i - \bar{x} )( y_i - \bar{y} )\right)^2 }{\sum ( y_i - \bar{y} ) ^2 \sum (x_i - \bar{x} ) ^2}\\ & = r^2. \end{align}